Tilburg University

Robust Control Charts for Time Series Data

Croux, C.; Gelper, S.; Mahieu, K.

Publication date:2010

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Citation for published version (APA):Croux, C., Gelper, S., & Mahieu, K. (2010). Robust Control Charts for Time Series Data. (CentER DiscussionPaper; Vol. 2010-107). Econometrics.

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Download date: 13. Dec. 2021

ROBUST CONT

By

ISSN 0924

No. 2010–107

ROBUST CONTROL CHARTS FOR TIME SERIES DATA

By Christophe Croux, Sarah Gelper and Koen Mahieu

October 2010

ISSN 0924-7815

ROL CHARTS FOR TIME SERIES DATA

Gelper and Koen Mahieu

Robust control charts for time series data

Christophe Croux

K.U. Leuven & Tilburg University

Sarah Gelper

Erasmus University Rotterdam

Koen Mahieu

K.U. Leuven

Abstract

This article presents a control chart for time series data, based on the one-step-

ahead forecast errors of the Holt-Winters forecasting method. We use robust

techniques to prevent that outliers affect the estimation of the control limits

of the chart. Moreover, robustness is important to maintain the reliability of

the control chart after the occurrence of alarm observations. The properties

of the new control chart are examined in a simulation study and on a real

data example.

Keywords: Control chart, Holt-Winters, Non-stationary time series, Out-

lier detection, Robustness, Statistical process control.

1 Introduction

Control charts are used to detect anomalies in processes. They are most often

used to monitor production-related processes. Samples taken from such processes

are assumed to be independent and identically distributed. Many business-related

processes, for instance sales volumes or product prices, behave very differently. They

typically contain a trend, local or global, and serial correlation. In this paper we

propose a control chart aimed at detecting aberrant observations, or outliers, in

such time series. The control chart needs to be robust, meaning that the presence

of outliers in the series should not harm its performance. Outliers may be present

in the “training period”, being the part of the series used to determine the control

limits, or in the “test period”, being the part of the series where outliers should be

flagged as alarm observations.

1

The control chart we propose belongs to the class of special-cause control charts

(Alwan and Roberts, 1988), where forecast errors of a prediction method are subject

to regular control chart techniques. If an observation has a large deviation from its

predicted value, an unexpected event occurred, and an alarm should be given. A

large difference between the observed and predicted value implies an unusually large

forecasting error, and the corresponding observation will be flagged as an outlier in

the control chart of the forecast errors. In this paper, we apply the Holt-Winters

forecast method, which is a widely used and simple procedure to forecast time series

containing trends and serial correlation. Moreover, it is known to yield good forecast

performance for many types of times series encountered in business and industry

(see Gardner (2006) and De Gooijer and Hyndman (2006) for recent applications).

Every single forecast error is then monitored on an X-chart for individual data.

Since the control chart should be resistant to outliers, a robust version of both the

Holt-Winters forecasting procedure and the X-chart needs to be used.

Several types of control charts for time series data have been proposed in the

literature. To deal with serial correlation in stationary processes ARMA-charts

(Jiang, Tsui, and Woodall, 2000) were proposed, or data transformations (Wang,

2005). CuScore charts (Box and Ramırez, 1992) can be of use for non-stationary

series; see Nembhard and Changpetch (2007) for a recent application. Vander Wiel

(1996) proposed control charts for processes that wander, based on integrated mov-

ing average models. The Holt-Winters method is an important example of this

approach. As the other proposals, the approach of Vander Wiel (1996) is sensitive

to outliers in the training sample and test sample. This problem is remediated by

following the robust approach taken in this paper.

Robust versions of the standard X and R chart are given in Rocke (1989), Rocke

(1992) and Tatum (1997), where the mean and scale of the process are estimated ro-

bustly for setting up the control limits. More recently, robust control charts for mul-

tivariate processes were proposed, see Vargas (2003), Alfaro and Ortega (2009), and

Stefatos and Hamza (2009). This paper contributes to this literature by providing a

robust control chart for time series typically encountered in business environments,

containing stochastic trends, outliers, and strong serial correlation.

In Section 2 control charts based on the Holt-Winters method are briefly ex-

plained, while Section 3 describes the robust version. The Holt-Winters method,

both the standard and the robust version, has the advantage of being easy to imple-

ment by means of a simple updating scheme. Section 4 presents a simulation study

that demonstrates the advantages of a robust approach. First, the robust version

2

gives reliable control limits in cases where the training sample contains outliers.

Secondly, if an isolated outlier occurs in the test sample, the non-robust control

chart might yield a sequence of false alarms right after the occurrence of the outlier.

The robust approach does not suffer from this drawback. Section 5 presents an

application with real data. Finally, some conclusions are made in Section 6.

2 Control charts using the Holt-Winters method

In this section, we construct a control chart for a time series {yt}. At every time

point we make a one-step-ahead forecast by the Holt-Winters forecasting algorithm.

The one-step-ahead forecast errors of this procedure are plotted on a control chart.

Control limits to monitor new incoming observations are constructed from the train-

ing sample.

2.1 The forecasting algorithm

Consider a time series observed up to time point t − 1. The Holt-Winters method

makes a prediction of the series at time t, denoted by yt|t−1. Once the true value yt

is observed, the one-step-ahead forecast error is given by

et = yt − yt|t−1. (1)

Denote the local level of the series by {αt} and the local trend by {βt}. The

Holt-Winters algorithm estimates these unknown values by

αt = λ1 yt + (1 − λ1) (αt−1 + βt−1)

βt = λ2 (αt − αt−1) + (1 − λ2) βt−1

(2)

resulting in the forecast

yt|t−1 = αt−1 + βt−1 . (3)

The parameters λ1 and λ2 in (2) are smoothing parameters, with values between

zero and one. The larger the smoothing parameters, the less the local level and/or

local trend series are smoothed, and the more weight the current value of yt has on

the prediction of the next value.

The recursive relations (2) are started up after a short startup period of length

m. A linear trend line is fitted to the data in the startup period for estimating the

3

local level and trend at t = m. The smoothing parameters λ1 and λ2 are selected

by minimizing the sum of squared forecast errors over a training period of length n:

(λopt

1 , λopt

2 ) = argmin(λ1,λ2)

n∑

t=m+1

(

yt − yt|t−1

)2. (4)

Note that the observations in the startup period are not used for determining the

smoothing parameters.

2.2 Control chart

As motivated in Section 1, the one-step-ahead forecast errors et = yt − yt|t−1 are

monitored to detect anomalies in the outcome of the process. We proceed as for a

regular X-chart, and assume normality of the forecast errors. The control limits are

computed from the forecast errors et in the training period, with t = m + 1, . . . , n.

Denote S2 the mean of the squared prediction errors in the training period, excluding

the startup values:

S2 =

∑nt=m+1 e

2t

n−m.

Since the target value for the forecast errors is zero, the following (1−α) Upper and

Lower Control Limits are obtained:

UCL = zα/2 × S

LCL = − zα/2 × S ,(5)

with zα/2 the (1 − α/2)-quantile of the standard normal distribution.

For every new observations yt in the test period, where t > n, one updates the

recursion relations (2) and computes the new prediction error et as in (1). The

value of et is then plotted in the control chart. If it falls out of the control limits,

then the observed value deviates substantially from the predicted value, indicating

an unexpected change in the process. Note that the observations in the test period

influence future prediction errors via the Holt-Winters forecast formulas in (2), but

do not alter the control limits.

It is of crucial importance that the process is completely in control during the

training period, otherwise the scale estimate S is inflated by the presence of outliers

in the training period. If outliers are present in the training sample, it is necessary

to follow a robust approach, as discussed in the next section.

4

3 Control charts using the Robust Holt-Winters

method

The control chart presented in the previous section is sensitive to outlying obser-

vations in several ways. First, the Holt-Winters forecasts are adversely affected by

outliers, since the estimated local level and trend in (2) depend on past values of the

observed series, and thus also on possible outliers. Second, if outliers are present in

the training sample, the selected smoothing parameters λ1 and λ2 might be biased.

Finally, due to an inflated scale of the forecast errors, outliers may lead to wider

control limits.

3.1 The forecasting algorithm

The shortcomings of the standard Holt-Winters algorithm have been addressed by

Gelper et al. (2009). They present a robust version of Holt-Winters method, consist-

ing of the application of the standard procedure on a cleaned version of the observed

series. Furthermore, they stress the importance of a robust selection of the smooth-

ing parameters and a robust estimation of the starting values of the algorithm. We

briefly review their approach.

A cleaned version y∗t of the time series yt is obtained as

y∗t = ψk

(

yt − αt−1 − βt−1

σt

)

σt + αt−1 + βt−1 (6)

where ψk(y) = max(−k,min(y, k)) is the Huber ψ-function with boundary value

k, and σt is a local scale estimate of the one-step-ahead forecast errors. We set

k = 2, such that the influence of forecast errors larger than two times the local

scale estimate is bounded. Note that for k tending to infinity, one gets the standard

Holt-Winters method again. The local scale estimate is computed recursively as

σ2t = λσρk

(

yt − αt−1 − βt−1

σt−1

)

σ2t−1 + (1 − λσ)σ2

t−1 (7)

where ρk(·) is a bounded loss function with boundary value k = 2, see Gelper et al.

(2009). Forecast errors larger than two times the local scale estimate σt−1 will have

a bounded influence on the next scale estimate. We allow for a slowly varying scale,

which is translated in a choice of λσ rather close to zero, for example λσ = 0.3.

5

The recursive scheme of the robust Holt-Winters algorithm then becomes, anal-

ogous to (2),

αt = λ1 y∗t + (1 − λ1)(αt−1 + βt−1)

βt = λ2 (αt − αt−1) + (1 − λ2) βt−1 ,(8)

where y∗t follows equation (6).

There are two further issues that need to be addressed. First, the startup values

αm and βm need to be computed using a robust regression fit to the data in the

startup period. For this robust fit the repeated median estimators is used. The

median absolute deviation of the residuals with respect to this robust fit yields the

starting value σ2m for the scale estimates. The second issue is the robust selection of

the smoothing parameters. Instead of minimizing the sum of squares of the one-step-

ahead forecast errors over the training sample, we minimize a robust performance

measure

(λopt

1 , λopt

2 ) = argmin(λ1,λ2)

s20

n∑

t=m+1

ρk

(

yt − yt|t−1

s0

)

. (9)

This performance measure corresponds to a τ -scale estimator, see for example

Maronna et al. (2006), which combines a high robustness with a high statistical effi-

ciency. The auxiliary scale estimate s0 in (9) is given by s0 = medm+1≤t≤n |yt−yt|t−1|,

the median absolute deviation of the prediction errors. The loss function used in

(9) is

ρk(y) = min(k2, y2) = ψk(y)2, (10)

with k = 2.

3.2 Control chart

Similar as in subsection 2.2, we plot the one-step-ahead forecast errors et = yt−yt|t−1,

for t > n, in the control chart, but now with forecasts obtained from the robust Holt-

Winters algorithm. The control limits are determined from a robust scale estimate

of the one-step-ahead forecast errors from the training period. We use again the

τ -estimator of scale

τ 2 = cks20

n−m

n∑

t=m+1

ρk

(

et

sT

)

, (11)

where s0 = medm+1≤t≤n |et|, and ρk as in (10). The consistency factor ck in the

above equation ensures that τ 2 is a consistent estimator of the population variance

of a normal distribution. For k = 2, we have ck = 1.404. We construct approximate

6

(1 − α) control limits as

UCL = zα/2 × τ

LCL = − zα/2 × τ(12)

with zα/2 as the (1 − α/2)-quantile of the standard normal distribution. Hence,

the formula for the control limits (5) and (12) are similar, except that the robust

method uses a robust τ -scale estimator instead of a standard deviation (centered at

zero). The use of the τ -scale estimator guarantees the resistance to outliers in the

training period.

In the next section a simulation study is carried out to verify whether the con-

structed control limits for the standard and the robust Holt-Winters method have

comparable properties if the training sample is clean. In the presence of outliers in

the training sample, we expect the robust method to perform better.

4 Simulation study

In this simulation study we take a closer look at the size, the power and the false

detection rate of the proposed robust control chart, and compare them to the same

characteristics of the non-robust version. The size of a control chart is the proba-

bility that an observation in the test sample falls out of the control limits when the

process is actually in control, i.e. when no outliers are present in the test sample.

It is supposed to be approximately 5%, when taking α = 0.05 in (5) and (12). The

power is the probability that an alarm observation is detected. In a simulation study,

the power equals the proportion of generated outliers in the test sample that are

detected. The percentage of observations in the test sample that are not generated

as outliers, but that are falsely identified as alarm observations, is the false detection

rate. The false detection rate is like the size of the control charts, but for a process

that is not fully under control.

4.1 Simulation design

We generate series {yt} from a local linear trend model. This model generates

non-stationary and correlated time series, and allows for trends. It is a reasonable

model for many series encountered in the practice of business and economics. At

this model the forecast errors of the Holt-Winters method form an i.i.d. sequence

7

from a normal distribution (e.g. Chatfield and Yar (1988)). The local linear trend

model is defined as

Yt = αt + εt , εt ∼ N(0, σ2ε) , (13)

where αt, the local level, and βt, the local trend, are given by

{

αt = αt−1 + βt−1 + ηt, ηt ∼ N(0, σ2η),

βt = βt−1 + νt, νt ∼ N(0, σ2ν). ,

(14)

The noise terms εt, ηt and νt are independent, and serially uncorrelated. In the

simulation study, we take σε = 1 and ση = σν = 0.1, leading to a smaller variance

for the local level and trend compared to the variance in the measurement equation

(13).

The first n observations from a generated series constitute the training sample,

including a startup period of length m = 10. The next 200 generated observations

constitute the test sample. From the training data the control limits are constructed

as outlined in Sections 2 and 3. Outliers, hence alarm observations, are induced in

the test sample by adding the value kσε to 10% of randomly selected observations in

the test sample. For simulating the power, we take k = 5, corresponding to outliers

of moderate size. To study the robustness of the size and power with respect to

outliers in the training sample, the outliers are generated in the same way as for the

test sample. The percentage of outliers in the training data are taken as 0% (clean),

2%, and 5%, respectively.

4.2 Simulation Results

The simulated power and size are summarized in Table 1 for the control charts based

on the standard and the robust Holt-Winters algorithm. The reported numbers are

averages over 1000 simulation runs. Training samples of length n = 50 and n = 100

are considered. The simulation study, although rather modest in the number of

considered simulation settings, is quite time consuming, since the optimal selection

of the smoothing parameters, see (4) and (9), needs to be performed for every single

time series.

As we see from Table 1 there is a small size distortion in the clean case, when

no outliers are present in the training sample, and this for both methods. For the

larger sample size, the simulated size is already much closer to 5%. In the presence

of outliers in the training sample, the size of the non-robust chart is close to zero

in most cases, while the size of the robust method is kept much more stable. The

8

Table 1: Size and Power of the control chart, with α = 5%, for the standard (HW)

and the robust Holt-Winters (RHW) procedures, for different sizes of the training

sample (50 and 100), and for different percentages of outliers in training sample.

Size Clean 2% 5%

n HW RHW HW RHW HW RHW

50 .073 .086 .044 .073 .026 .067

100 .059 .063 .027 .052 .009 .037

Power Clean 2% 5%

n HW RHW HW RHW HW RHW

50 .903 .900 .843 .874 .784 .850

100 .903 .902 .842 .881 .757 .853

breakdown of the size for the non-robust method is due to two reasons. The main

reason is that the scale S in (5) is inflated due to the outliers, resulting in too

wide control limits. Furthermore, the outliers cause a bias in the selection of the

smoothing parameters, leading to a less optimal smoothing.

Consider now the power in Table 1. If the training sample is clean, the two meth-

ods have a comparable power to detect the generated alarm observations. When 2%

or 5% of outliers are present in the training sample, they do affect the detection

power of the control charts, and we see that the robust Holt-Winters clearly out-

performs the standard one. The difference in power is about 10% in presence of

only 5% of moderate outliers in the training sample. When increasing k, the size of

the outliers, and the percentage of outliers in the training sample, these differences

become even much more pronounced.

Next, we compare the false detection rate of the two control charts. We follow the

same simulation scheme as before, but only consider clean training data to ensure

that the size of the two methods remains comparable. We vary the magnitude of

the outliers by taking k = 5, 10, 15 and 20. A first observation is that the false

detection rate does not seem to depend much on n. Furthermore, for k = 5 the

robust method is only slightly better than the non-robust method. For larger values

of k, in contrast, the robust procedure has a much lower false detection rate. This

is due to the following reason. For large values of k, an outlying observation in the

test sample affects the outcome of the next forecast in the Holt-Winters procedure.

9

Table 2: False detection rate of the control chart, with α = 5%, for the standard

(HW) and the robust Holt-Winters (RHW) algorithm, and for different lengths n

of the training sample. The training data are clean, and outliers are created in the

test data by shifting 10% of the observations upwards by k units.

HW RHW

shift size k = 5 k = 10 k = 15 k = 20 k = 5 k = 10 k = 15 k = 20

n = 50 0.097 0.167 0.206 0.232 0.084 0.088 0.083 0.085

n = 100 0.093 0.161 0.203 0.229 0.079 0.080 0.081 0.082

Hence it is likely to induce a large forecasting error for the next observation as well,

even if the latter one is not an outlier. The propagation effect of outliers does not

occur with the robust approach, since it uses “cleaned observations” in the update

formulas (8).

5 Real data example

We illustrate the proposed method by means of a real data example. We use the

Housing Data from the book of Diebold (2001), containing monthly data for housing

starts and completions from January 1968 until June 1996. Figure 1 shows both

series and the corresponding control charts obtained by the standard and the robust

Holt-Winters procedure. The upper panel of every graph shows the observed data yt,

together with the smoothed series yt|t−1. The lower panel of every graph contains the

control chart. As explained in section 2, the prediction errors et are plotted in the

control chart. The vertical dashed lines in the chart indicate the end of the startup

period and of the training period. Control limits are indicated for the test period.

We used a startup period of length m = 20, a training period of length n = 150.

The other 192 observations serve as the test sample. We immediately notice two

large outliers, one near 1971 and another near 1977, in the training sample of the

housing starts. Moreover, both time series may contain other smaller outliers. By

applying robust exponential smoothing, we expect that the results remain stable in

the presence of these outliers.

We first take a closer look at the housing starts in Figure 1(a)-(b). The smoothing

parameters (λ1, λ2) selected from the training period are (0.3, 0.2) for the standard

10

(a)

0.5

1.5

2.5

Standard H−W

Hou

sing

sta

rts

1970 1975 1980 1985 1990 1995

−2.

0−

1.0

0.0

1.0

fore

cast

err

ors

(b)

0.5

1.5

2.5

Robust H−W

Hou

sing

sta

rts

1970 1975 1980 1985 1990 1995

−2.

0−

1.0

0.0

1.0

fore

cast

err

ors

(c)

0.5

1.5

2.5

Standard H−W

Hou

sing

com

plet

ions

1970 1975 1980 1985 1990 1995

−0.

40.

00.

20.

4

fore

cast

err

ors

(d)

0.5

1.5

2.5

Robust H−W

Hou

sing

com

plet

ions

1970 1975 1980 1985 1990 1995

−0.

40.

00.

20.

4

fore

cast

err

ors

Figure 1: In each graph, the upper panel presents the observed data (dots) and the smoothed

series (solid line), while the lower panel presents the control chart, with forecast errors (points)

and control limits (dashed lines). The housing starts series - containing outliers in the training

period - is analyzed with standard Holt-Winters in (a) and with robust Holt-Winters method in

(b). Similarly for the housing completions series. This series does not contain outliers in the

training period, explaining the resemblance between (c) and (d).

11

method and (0.4, 0.2) for the robust method. These parameters are reasonable,

since they are close to zero and the parameter corresponding to the trend is smaller

than the one for the level. The standard method tends to select smaller parameters

than the robust version, since outliers may bias the parameters towards zero. The

direct effect of the large outliers in the standard Holt-Winters case can be seen

at the time points subsequent to the large outliers. The forecasts for these time

points are attracted by the outlier and therefore, in this case, the values are too

small. The robust method clearly yields better forecasts in the outlier’s subsequent

time points. Also in terms of mean squared one-step-ahead forecast errors (MSFE),

computed over the test period, the robust method outperforms the standard one

with an MSFE of 0.0144 against 0.0175. The control chart shows that the control

limits are inflated by the outliers in the non-robust case, and all observations in

the test sample remain within the control bounds. This leads to a loss of power of

the non-robust method to detect outliers. For example, the alarm observation in

January 1984 is only detected by the robust method.

The housing completions in Figure 1(a)-(b), do not show important outliers in

the training period. The selected smoothing parameters for this series are (0.5, 0.2)

in the standard case and (0.6, 0.3) in the robust case. Similar to the housing starts,

the standard parameters are somewhat smaller than the robust ones. The MSFE for

both methods are comparable, with 6.01×10−3 in the standard case and 6.09×10−3

in the robust case. Since the series do not contain important outliers, both the

standard and the robust control charts look similar. Nevertheless, the robust control

chart has slightly tighter control limits. The percentage of observations exceeding

the control limits is 2% for the standard method, which seems overly optimistic, and

6.1% for the robust method, closer to the expected size of α = 5%.

We conclude that it is worthwhile to use the robust method. If there are no

major outliers, the performance of both methods is quite similar. However, if major

outliers are present, in particular in the training period, the robust method is more

reliable.

6 Conclusion

In this paper, we propose to use the one-step-ahead forecast errors of the Holt-

Winters forecasting method as the input of a control chart for monitoring non-

stationary processes. Holt-Winters forecasting is a frequently used method for uni-

variate time series with a stochastic trend and a seasonality pattern. In this study,

12

we did not explicitly deal with seasonality, but the method can easily be adapted for

that purpose. The main advantages are that the method is very easy to implement

and still performs well compared to other more complicated forecasting techniques.

We compared the control charts based on the standard Holt-Winters method with

a robust version.

The simulation study shows that with clean training data, the standard and

robust control charts perform almost equally well in terms of size and power. On the

other hand, when we add a small amount of moderate outliers to the training sample

the non robust method looses power, and suffers from severe size distortion. Even

when the training data are clean, the false detection rate of the robust procedures is

much lower if major outliers are present in the test sample. Finally, the application

to real data shows that the proposed robust method is easy to put in practice. The

graphical displays in Figure 1, representing both the observed series and the control

chart in the same graph, are an easy-to-use visual aid for managers to monitor

business processes.

There are several open questions, which we consider as areas for future research.

If a structural break appears in the time series, the control chart should be readjusted

and new control limits need to be computed. Another limitation of the presented

approach is the normality assumption on the forecast errors (excluding outliers).

Different distributional assumptions will lead to different formula for the control

limits. We refer to Yang and Chen (2009) and Chen and Yeh (2009) for recent

contributions to these problems, although in different settings.

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